Solid State TheoryExercise 2FS 11Prof. M. SigristPoint groups and their representationsExercise 2.1Energy bands of almost free electrons on the fcc latticeLet us consider almost free electrons on a face-centered cubic (fcc) lattice. The goal ofthis exercise is to compute the lowest energy bands along the Δ-line using degenerateperturbation theory and the machinery ofgroup theory.Remember that in reciprocalspace, the fcc lattice transforms into a body-centered cubic (bcc) lattice. The point groupof the cubic Bravais lattices (simple cubic, fcc, bcc) is denoted byOh(symmetry groupof a cube). Its character table is given in Tab. 1.a) We first study the Γ point (~k= 0). Forfreeelectrons (V= 0) the lowest energy levelis non-degenerate and the second one has an eight fold degeneracy. We focus on thesecond level and denote the eight-dimensional representation ofOhdefined on thissubspace by Γ. Find the irreducible representations contained in Γ. Compute thegroup characterχΓ

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